m-adic residue codes

Abstract

The m-adic residue codes are a generalization of the quadratic residue codes. They are cyclic codes which exist at prime lengths p over GF(q) when m I(pl), ( q , p ) = 1, and q is an m-adic residue modulo p . The m-adic residue codes are investigated and are found to have many of the strong properties of the quadratic residue codes. A subgroup of the automorphism group and restrictions on the form of the idempotents of the m-adic residue codes are given. It is shown that some m-adic residue codes are self-orthogonal and the duals of some m-adic residue codes are their complements. Bounds on the minimum of the weights of the odd-like vectors in the odd-like codes are given. (For binary codes, this is the minimum odd weight.) At some lengths, m-adic residue codes exist for several values of m. Containment relationships between these codes are demonstrated which show that when m is even, m-adic residue codes inherit properties of quadratic residue codes. A table is included that contains minimum weights of the binary m-adic residue codes of lengths less than or equal to 127. Many of these codes have the highest possible minimum weight known for codes of their lengths and dimensions. Index Terms-Cyclic codes, quadratic residue codes, m-adic residue codes.

Polyadic Cyclic Codes over a Non-Chain Ring

“…, then polyadic codes are m-adic residue codes as defined by Job [21]. A polyadic code of prime length p exists if and only if q ia an m-adic residue mod p, see Brualdi and Pless [22].…”

New error correcting codes for informed receivers

“…In summary, C 1 , C 2 and C 3 are all [31, 10,12] codes, the sum of any two of these three codes is [31, 20,6], and the sum of all three is the [31, 30, 2] code. All these codes have the optimum minimum distance for their length and dimension.…”

“…Since T 1 and T 2 form a partition of Z * n , {C 1 , C 2 } is a binary ECCIR with C 1 + C 2 being the singleparity check code. The QR codes C 1 , C 2 for the first few values of n are [7,3,4], [17,8,6], [23, 11,8], [31,15,8], [41,20,10], [47,23,12]. For each of the corresponding ECCIRs, C 1 , C 2 and C 1 + C 2 have the optimum minimum distances.…”

“…The scheme of Section V-D can be extended to L = 3 using cubic residue (CR) codes [20], [21]. Binary CR codes are defined for all prime lengths n for which 3|(n − 1) and 2 is a cubic residue mod n. The set of cubic residues mod n form a subgroup of index 3 in Z * n .…”

Polyadic codes of prime power length

“…When X ∞ = {0}, N = p an odd prime and X 0 is the set of all m-adic residues modulo p, polyadic codes are m-adic residue codes defined by Brualdi and Pless [3], and investigated further, by V.R. Job [8]. Ding, Kohel and Ling [5] generalized duadic codes to split group codes, where the underlying group was taken to be any finite Abelian group.…”